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Electronic Colloquium on Computational Complexity

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TR14-117 | 18th August 2014 11:58

Fast Approximate Matrix Multiplication by Solving Linear Systems



In this paper, we present novel deterministic algorithms for multiplying two $n \times n$ matrices approximately. Given two matrices $A,B$ we return a matrix $C'$ which is an \emph{approximation} to $C = AB$. We consider the notion of approximate matrix multiplication in which the objective is to make the Frobenius norm of the error matrix $C-C'$ arbitrarily small. Our main contribution is to first reduce the matrix multiplication problem to solving a set of linear equations and then use standard techniques to find an approximate solution to that system in $\tilde{O}(n^2)$ time. To the best of our knowledge this the first examination into designing quadratic time deterministic algorithms for approximate matrix multiplication which guarantee arbitrarily low \emph{absolute error} w.r.t. Frobenius norm.


Comment #1 to TR14-117 | 23rd February 2015 14:04

Fast Approximate Matrix Multiplication in ECCC TR14-117 Fails to Converge

Comment #1
Authors: Kevin Kowalski, Dieter van Melkebeek
Accepted on: 23rd February 2015 14:04
Downloads: 1475


ECCC TR14-117 presents an efficient iterative algorithm that purportedly computes an approximation to the product of two $n \times n$ matrices. On this comment we show that, in fact, the algorithm converges to a diff erent matrix than the product matrix.

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